The Kramers–Kronig relations, sometimes abbreviated as KK relations, are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. The relations are often used to compute the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the condition of analyticity, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hans Kramers. In mathematics, these relations are known by the names Sokhotski–Plemelj theorem and Hilbert transform.
Formulation
Let χ ( ω ) = χ 1 ( ω ) + i χ 2 ( ω ) {\displaystyle \chi (\omega )=\chi _{1}(\omega )+i\chi _{2}(\omega )} be a complex function of the complex variable ω {\displaystyle \omega } , where χ 1 ( ω ) {\displaystyle \chi _{1}(\omega )} and χ 2 ( ω ) {\displaystyle \chi _{2}(\omega )} are real. Suppose this function is analytic in the closed upper half-plane of ω {\displaystyle \omega } and tends to 0 {\displaystyle 0} as | ω | → ∞ {\displaystyle |\omega |\to \infty } . The Kramers–Kronig relations are given by
χ 1 ( ω ) = 1 π P ∫ − ∞ ∞ χ 2 ( ω ′ ) ω ′ − ω d ω ′ {\displaystyle \chi _{1}(\omega )={\frac {1}{\pi }}{\mathcal {P}}\!\!\int _{-\infty }^{\infty }{\frac {\chi _{2}(\omega ')}{\omega '-\omega }}\,d\omega '}
and
χ 2 ( ω ) = − 1 π P ∫ − ∞ ∞ χ 1 ( ω ′ ) ω ′ − ω d ω ′ , {\displaystyle \chi _{2}(\omega )=-{\frac {1}{\pi }}{\mathcal {P}}\!\!\int _{-\infty }^{\infty }{\frac {\chi _{1}(\omega ')}{\omega '-\omega }}\,d\omega ',}
where ω {\displaystyle \omega } is real and where P {\displaystyle {\mathcal {P}}} denotes the Cauchy principal value. The real and imaginary parts of such a function are not independent, allowing the full function to be reconstructed given just one of its parts.
Derivation
The proof begins with an application of Cauchy's residue theorem for complex integration. Given any analytic function χ {\displaystyle \chi } in the closed upper half-plane, the function ω ′ ↦ χ ( ω ′ ) / ( ω ′ − ω ) {\displaystyle \omega '\mapsto \chi (\omega ')/(\omega '-\omega )} , where ω {\displaystyle \omega } is real, is analytic in the (open) upper half-plane. The residue theorem consequently states that
∮ χ ( ω ′ ) ω ′ − ω d ω ′ = 0 {\displaystyle \oint {\frac {\chi (\omega ')}{\omega '-\omega }}\,d\omega '=0}
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